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Mathematics 5.

Table of contents
Natural numbers
Geometry
Measurement, statistics
Angles
Fractions
The rectangle
The cuboid
Decimals
Integers
Orientation
Orientation in the world around you
Orientation in the plane
Graphs
Mixed exercises
Appendix
Orientation in the plane
Orientation on a line is simple because one primary piece of information is enough. On a number line, one point corresponds to one number.
\latex{ -6 }
\latex{ -5 }
\latex{ -4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
Orientation on a plane, however, requires two pieces of information. It only becomes possible if you put two perpendicular number lines together so that they cross at a \latex{ 0 } point.
The coordinate system consists of two perpendicular lines at right angles.
• These lines are called the \latex{ x }-axis and the \latex{ y }-axis.
• The intersection of the two axes is called the origin.
• The two number lines divide the plane into four quadrants.
\latex{ Quadrant\; IV }
\latex{ Quadrant\; I }
\latex{ Quadrant\; II }
\latex{ Quadrant\; III }
\latex{ origin }
\latex{ y-axis }
\latex{ x-axis }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 0 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ -5 }
\latex{ -6 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ -5 }
The coordinate system is also known as the Cartesian plane or the coordinate or number plane.
Example 1
What are the coordinates of point \latex{ P } on the Cartesian plane?
Solution
The line passing through point \latex{ P } and perpendicular to the \latex{ x }-axis intersects the \latex{ x }-axis at \latex{ -3 }, which is the first coordinate of point \latex{ P }.

Similarly, the line passing through point \latex{ P } and perpendicular to the \latex{ y }-axis intersects the \latex{ y }-axis at \latex{ 4 }, which is the second coordinate of point \latex{ P }.
\latex{P}
\latex{y}
\latex{x}
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 0 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ -5 }
\latex{ -6 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ -1 }
\latex{ -2 }
René Descartes
(\latex{ 1596–1650 })
French philosopher, scientist and mathematician. He used the coordinate system to translate mathematical problems into geometric ones.
The first coordinate of point \latex{ P } is \latex{ -3 }, and the second is \latex{ 4 }. In short: \latex{ P } (\latex{ -3}; \latex{ 4 } ).
650401
  • The first coordinate of point \latex{ A } is \latex{ 2 }, and the second is \latex{ 4 }. In short: \latex{ A } \latex{ (2; 4) }.
  • The first coordinate of point \latex{ B } is \latex{ 4 }, and the second is \latex{ 2 }. In short: \latex{ B } \latex{ (4; 2) }.
  • In the case of \latex{ (2; 4) } and \latex{ (4; 2) }, the order of the numbers is different, so they define different points on the coordinate system.
 
The coordinate system assigns a pair of numbers to each point on the plane (two numbers in a defined order).
Each ordered pair of numbers defines a point on the plane. (For example, \latex{ (0; 0) } defines the origin.)
\latex{A}
(
\latex{ 2 }
;
\latex{ 4 }
)
\latex{B}
(
\latex{ 4 }
;
\latex{ 2 }
)
\latex{y}
\latex{x}
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 0 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
Example 2
Mark point \latex{ P } on the Cartesian plane based on its coordinates \latex{ (-5; -2) }.
Solution
The first coordinate of point \latex{ P } is \latex{ -5 }.
The line perpendicular to the \latex{ x }-axis through point \latex{ -5 } contains all the numbers whose first coordinate is \latex{ -5 }.
The second coordinate of point \latex{ P } is \latex{ -2 }. The line perpendicular to the \latex{ y }-axis at point \latex{ -2 } contains all the numbers whose second coordinate is \latex{ -2 }.
Point \latex{ P } \latex{ (-5; -2) } is on both lines, so it will be at the intersection of the two lines.
\latex{P}
(
\latex{ -5}
;
\latex{ -2 }
)
\latex{y}
\latex{x}
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 0 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -5 }
\latex{ -6 }
\latex{ -7 }
\latex{ -8 }
\latex{ -4 }
Exercises
657987
{{exercise_number}}. Write the coordinates of each of the points shown on the coordinate system.
\latex{ -7 }
\latex{ -6 }
\latex{ -5 }
\latex{ -4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ -5 }
\latex{y}
\latex{x}
\latex{A}
\latex{B}
\latex{C}
\latex{D}
\latex{E}
\latex{F}
\latex{G}
\latex{H}
\latex{ -7 }
\latex{ -6 }
\latex{ -5 }
\latex{ -4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ -5 }
\latex{y}
\latex{x}
\latex{A}
\latex{B}
\latex{C}
\latex{D}
\latex{E}
\latex{F}
\latex{G}
\latex{H}
{{exercise_number}}. Mark \latex{ 3 } arbitrary points on each axis of the coordinate system and write down their coordinates. What do the points on each axis have in common?
{{exercise_number}}. Mark \latex{ 3 } arbitrary points in each quadrant and write down their coordinates. What do the points in each quadrant have in common?
{{exercise_number}}. What are the coordinates of the points on the pine tree? (→)
\latex{ -7 }
\latex{ -6 }
\latex{ -5 }
\latex{ -4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
 \latex{H}
\latex{F}
\latex{G}
\latex{E}
\latex{D}
\latex{L}
\latex{K}
\latex{M}
\latex{N}
\latex{O}
\latex{A}
\latex{B}
\latex{C}
\latex{x}
\latex{y}
\latex{I}
\latex{J}
{{exercise_number}}. Mark \latex{ 3 } arbitrary points on a coordinate system starting with the following coordinates:
a) \latex{ 3 }
b) \latex{ 7 }
c) \latex{ -2 }
d) \latex{ -5 }
e) \latex{ 1 }
f) \latex{ 0 }
Where are the points located?
{{exercise_number}}. Mark \latex{ 3 } arbitrary points on a coordinate system with the following second coordinates:
657695
a) \latex{ 2 }
b) \latex{ 6 }
c) \latex{ -3 }
d) \latex{ -7 }
e) \latex{ -1 }
f) \latex{ 0 }
Where are the points located?
{{exercise_number}}. Mark the following points on a coordinate system.
a) A\latex{ (3; 5) }
b) B\latex{ (-3; 5) }
c) C\latex{ (4; -2) }
d) D\latex{ (-3; -5) }
e) E\latex{ (3; -5) }
f) F\latex{ (-5; -4) }
g) G\latex{ (0; 2) }
h) H\latex{ (-3; 0) }
{{exercise_number}}. Mark the following points on a coordinate system.
a) A\latex{ (1; 2) }
b) B\latex{ (-3; -3) }
c) C\latex{ (2; -3) }
d) D\latex{ (-4; 1) }
e) E\latex{ (4; 0) }
f) F\latex{ (-2; -3) }
g) G\latex{ (0; -5) }
h) H\latex{ (0; 0) }
{{exercise_number}}. Which quadrants are the following points located in?
a) A\latex{ (3; 2) }
b) B\latex{ (-4; 5) }
c) C\latex{ (-4; -3) }
d) D\latex{ (5; -2) }
e) E\latex{ (16; 25) }
f) F\latex{ (-1,000; 1,000) }
g) G\latex{ (1,000; -1,000) }
h) H\latex{ (-1,000; -1,000) }
{{exercise_number}}. Mark the following points on a coordinate system, then connect them in alphabetical order. What do you get?
A\latex{ (0; 0) }
B\latex{ (3; 0) }
C\latex{ (4; -1) }
D\latex{ (5; -1) }
E\latex{ (6; 0) }
F\latex{ (7; 0) }
G\latex{ (7; 3) }
H\latex{ (5; 5) }
I\latex{ (-1; 5) }
J\latex{ (-2; 3) }
K\latex{ (-5; 2) }
L\latex{ (-5; 0) }
M\latex{ (-4; 0) }
N\latex{ (-3; -1) }
P\latex{ (-2; -1) }
Q\latex{ (-1; 0) }
Quiz
The Cartesian spider, a mythical species, can crawl one unit at a time on a coordinate system. Starting from the origin, it first crawls along an axis, then moves again perpendicular to its previous direction. After that, it takes another perpendicular step. Mark all the possible coordinates where the spider could have ended its journey.
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{ -4 }
\latex{ -5 }
\latex{ -1 }
\latex{ -2 }
\latex{ -3 }
\latex{y}
\latex{x}
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